Reconstructing Bounded Treelength Graphs with Linearithmic Shortest Path Distance Queries

TL;DR

This paper presents a deterministic algorithm for reconstructing bounded degree and treelength graphs using a near-optimal number of shortest path queries, improving previous methods by a logarithmic factor.

Contribution

It introduces an $O(n ext{ log } n)$ query algorithm for reconstructing bounded treelength graphs, matching lower bounds and improving over prior algorithms.

Findings

Reconstruction algorithm uses $O_{ ext{Δ,tl}}(n ext{ log } n)$ queries.

Algorithm is deterministic and optimal for bounded treelength graphs.

Improves previous bounds by a $ ext{log } n$ factor.

Abstract

We consider the following graph reconstruction problem: given an unweighted connected graph $G = (V,E)$ with visible vertex set $V$ and an oracle which takes two vertices $u,v \in V$ and returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? Specifically, we consider bounded degree $\Delta$ and bounded treelength $\mathrm{tl}$ connected graphs and show that reconstruction can be done in $O_{\Delta,\mathrm{tl}}(n \log n)$ queries with a deterministic algorithm. This result improves over the best known algorithm (deterministic or randomized) for this graph class by a $\log n$ factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.